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User:Temperal/The Problem Solver's Resource7: Difference between revisions

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{{User:Temperal/testtemplate|page 7}}
<br /><br />
==<span style="font-size:20px; color: blue;">Limits</span>==
{| style='background:lime;border-width: 5px;border-color: limegreen;border-style: outset;opacity: 0.8;width:840px;height:300px;position:relative;top:10px;'
This section covers limits and some other precalculus topics.
|+ <span style="background:aqua; border:1px solid black; opacity: 0.6;font-size:30px;position:relative;bottom:8px;border-width: 5px;border-color:blue;border-style: groove;position:absolute;top:50px;right:155px;width:820px;height:40px;padding:5px;">The Problem Solver's Resource</span>
===Definition===
|-
 
| style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 7}}
*<math>\lim_{x\to n}f(x)</math> is the value that <math>f(x)</math> approaches as <math>x</math> approaches <math>n</math>.
==<span style="font-size:20px; color: blue;">Combinatorics</span>==
 
<!-- will fill in later! -->
*<math>\lim_{x\uparrow n}f(x)</math> is the value that <math>f(x)</math> approaches as <math>x</math> approaches <math>n</math> from values of <math>x</math> less than <math>n</math>.
==Factorial==
 
The factorial of a number <math>n</math> is <math>n(n-1)(n-2)...(1)</math> or also as <math>\prod_{a=0}^{n-1}(n-a)</math>,and is denoted by <math>n!</math>.
*<math>\lim_{x\downarrow n}f(x)</math> is the value that <math>f(x)</math> approaches as <math>x</math> approaches <math>n</math> from values of <math>x</math> more than <math>n</math>.
 
*If <math>\lim_{x\to n}f(x)=f(n)</math>, then <math>f(x)</math> is said to be continuous in <math>n</math>.
 
===Properties===
 
Let <math>f</math> and <math>g</math> be real functions. Then:
*<math>\lim(f+g)(x)=\lim f(x)+\lim g(x)</math>
*<math>\lim(f-g)(x)=\lim f(x)-\lim g(x)</math>
*<math>\lim(f\cdot g)(x)=\lim f(x)\cdot\lim g(x)</math>
*<math>\lim\left(\frac{f}{g}\right)(x)=\frac{\lim f(x)}{\lim g(x)}</math>
 
===Squeeze Play Theorem (or Sandwich Theorem)===
Suppose <math>f(x)</math> is between <math>g(x)</math> and <math>h(x)</math> for all <math>x</math> in the neighborhood of <math>S</math>. If <math>g</math> and <math>h</math> approach some common limit L as <math>x</math> approaches <math>S</math>, then <math>\lim_{x\to S}f(x)=L</math>.
 
 
===Diverging-Converging Theorem===
A series <math>\sum_{i=0}^{\infty}S_i</math> converges iff <math>\lim S_i=0</math>.
 
===Focus Theorem===
The statement <math>\lim_{x\to n}f(x)=L</math> is equivalent to: given a positive number <math>\epsilon</math>, there is a positive number <math>\gamma</math> such that <math>0<|x-n|<\gamma\Rightarrow |f(x)-L|<\epsilon</math>.


Also, <math>0!=1</math>.


[[User:Temperal/The Problem Solver's Resource6|Back to page 6]] | [[User:Temperal/The Problem Solver's Resource8|Continue to page 8]]
[[User:Temperal/The Problem Solver's Resource6|Back to page 6]] | [[User:Temperal/The Problem Solver's Resource8|Continue to page 8]]
|}<br /><br />

Latest revision as of 09:59, 19 June 2025


Introduction Other Tips and Tricks Methods of Proof You are currently viewing page 7.

Limits

This section covers limits and some other precalculus topics.

Definition

  • $\lim_{x\to n}f(x)$ is the value that $f(x)$ approaches as $x$ approaches $n$.
  • $\lim_{x\uparrow n}f(x)$ is the value that $f(x)$ approaches as $x$ approaches $n$ from values of $x$ less than $n$.
  • $\lim_{x\downarrow n}f(x)$ is the value that $f(x)$ approaches as $x$ approaches $n$ from values of $x$ more than $n$.
  • If $\lim_{x\to n}f(x)=f(n)$, then $f(x)$ is said to be continuous in $n$.

Properties

Let $f$ and $g$ be real functions. Then:

  • $\lim(f+g)(x)=\lim f(x)+\lim g(x)$
  • $\lim(f-g)(x)=\lim f(x)-\lim g(x)$
  • $\lim(f\cdot g)(x)=\lim f(x)\cdot\lim g(x)$
  • $\lim\left(\frac{f}{g}\right)(x)=\frac{\lim f(x)}{\lim g(x)}$

Squeeze Play Theorem (or Sandwich Theorem)

Suppose $f(x)$ is between $g(x)$ and $h(x)$ for all $x$ in the neighborhood of $S$. If $g$ and $h$ approach some common limit L as $x$ approaches $S$, then $\lim_{x\to S}f(x)=L$.


Diverging-Converging Theorem

A series $\sum_{i=0}^{\infty}S_i$ converges iff $\lim S_i=0$.

Focus Theorem

The statement $\lim_{x\to n}f(x)=L$ is equivalent to: given a positive number $\epsilon$, there is a positive number $\gamma$ such that $0<|x-n|<\gamma\Rightarrow |f(x)-L|<\epsilon$.


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